Update on the chair: it's finished!
It took a fair amount of elbow grease (and many, many wasted staples) but I got there in the end, thanks to a lot of help from Matt. We had a struggle with the back piece and couldn't get the plastic back off. So we ended up stapling as close as we could to the plastic shell and tucking it all underneath. As a result I was expecting it to end up very messy - but was pleasantly surprised. It won't win any beauty contests or any prizes for perfection, but it's mine and I love it.